The Phase Plane

The Phase Plane
A Tool for Gaining Insight Into Nonlinear System Behavior

Why Use The Phase Plane?
Limit Cycles

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Why Use The Phase Plane?

The phase plane is a tool that will let you get a different viewpoint about some of the phenomena that can be found in nonlinear systems. There are a number of phenomena that are peculiar to nonlinear systems, and the phase plane can help you to understand and visualize what might happen in a nonlinear system. Don't underestimate the usefullness of visualizing things. Engineers communicate pictorially more than they do verbally. Electrical engineers use circuit diagrams to communicate, for example, and other branches of engineering use their own kinds of diagrams to communicate, from blueprints for construction engineers to flow diagrams for software engineers.

Here are some of the kinds of things that you will want to understand about nonlinear systems.

Linear systems might oscillate at a particular frequency, but oscillation at a particular amplitude is a phenomenon peculiar to nonlinear systems. That phenomenon is called a limit cycle.
Nonlinear systems can exhibit instability when certain inputs are applied but may be well behaved (stable) for other inputs.
There are very few analytical techniques that can be used to predict behavior of nonlinear systems.

For comparison, many universities teach courses in Linear Systems, and in those courses general techniques for analysis of linear systems are taught.
General techniques for analysis of nonlinear systems are hard or impossible to find and we are often left to use very specific techniques for special situations and we are not able to make general statements.

To make predictions of nonlinear system behavior designers often use simulations. However, doing a simulation of a nonlinear system only tells the designer about that situation. There could be other situations in which the system misbehaves, and the designer will only find that out if s/he does a simulation for that specific situation.

The Phase Plane - Introduction

The phase plane is restricted to second order systems. Second order systems have two state variables, and a phase plane is simply a plot of the two state variables against each other. It is called a phase plane because state variables were called phase variables back in the old days. (Actually, phase variables are a particular choice of state variables, but the term "phase plane" has come to encompass any plot of this sort.

We will start by examining a particular linear second order system. The system we will start working with has a transfer function as given below.

G(s) = Y(s)/U(s)
G(s) = 1/(s² + 2zs + 1)

This system is a second order system with the following parameters.

The natural frequency, w_n = 1.0
The damping ratio, z, is variable.
The DC gain is 10

The system can be described with two state equations.

First, let the output be a state, x₁. Then, we have the following.
The derivative of the output is a second state

dx₁/dt = x₂

Using the differential equation that relates input and output we can get a second state equation.

dx2/dt = - x₁ - 2zx₂ + u
The x's are the states, and u is the input.

We will examine the impulse response of a system.

The system starts from a non-zero initial state, with x₁ = 0.0 and x₂ = 10.0.

You can experiment with different damping ratios in this system, and then observe the behavior. Note the following.

The system is set initially for a damping ratio of 0.25. You can type a different value in the text box, then click the start button.
After the presentation has run its' course, clicking the start button once removes the plot, and clicking it a second time starts a new presentation.

You might not expect the behavior that you see. Clearly, both states oscillate and decay to zero. However, you might want to think about the following points.

If the system started from another state, the behavior would be pretty much the same.

For example, if the system started from x₁ = 0.0 and x₂ = -1.9, you would see the same behavior.

In essence, there are no special cases here. In theory, you could expand the scales, and you would see the same thing happening - the same spiral in to the origin.

There is insight that can be gained from the phase plane plot - insight that you might not get from a plot of either state against time.

One point to note is how the system behaves when the damping ratio is zero. Here's the simulation again. Use a value of zero for the damping ratio and note the result.

Notice how the curve closes on itself. That's and example of a limit cycle. In a linear system, the limit cycle can have any amplitude. Here is another version of the simulation above. In this one you can set the impulse magnitude and check the effects.

Question

Q1 Is it possible to have limit cycles of different amplitudes in this system?

Check your answer by using a value of zero for the damping ratio and using different amplitudes (5, 10 and 20).

Limit Cycles

Limit cycles in nonlinear systems are different than what you find in linear systems. In many nonlinear systems limit cycles may not have variable amplitude, and you may end up at the exact same limit cycle from many different starting points. To help you see that better, we are going to consider a simple nonlinear system. Actually, this system is well known, and it is called the Van der Pol system. It's not far removed from a linear system and we will use the example to lead you from a linear system to the Van der Pol system.

Example - The Van der Pol Oscillator

The Van der Pol oscillator is a second order system. It is a second order system where the damping term is nonlinear. Let's look at the state equations for a linear second order system again. Here they are again.

dx₁(t)/dt = x₂(t)
dx₂(t)/dt = - 2zw_nx₂(t) - w_n²x₁(t)+ w_n²G_dcu(t)

The damping ratio, z, appears in only one place in the second state equation. If a linear system is an oscillator the damping ratio would be zero. Actually, to get the oscillations started the damping ratio should be negative so that the oscillations will grow. In a purely linear system they would continue to grow forever. Let's consider the idea of putting something in the damping term that makes the damping negative (so the oscillations grow at the start) when the states are small, but makes the damping become positive if the states are large. Also, we won't have any input in the oscillator, so we will remove the input term. Let's summarize what we are going to do.

Let u(t) = 0
Set the undamped natural frequency to 1.0 to get a numerical example.
Fix the damping so that it becomes positive for larger states.

The way this is done in the Van der Pol oscillator is to use state equations as shown below.

dx₁(t)/dt = x₂(t)
dx₂(t)/dt = - 2z(1 - x₁²(t))x₂(t) - x₁ (t)

The addition of the (1 - x₁²(t)) term modifies the damping, and if the damping ratio is negative - for growing oscillations - as the oscillations grow and x₁(t) gets larger - the effective damping will become positive, preventing the oscillations from growing.

Actually, if you go into the literature you will find that the damping term is represented slightly differently. Here is the way you will find this system represented.

dx₁(t)/dt = x₂(t)
dx₂(t)/dt = e(1 - x₁²(t))x₂(t) - x₁ (t)

Here, the term - 2z (which is negative) as been replaced with a positive term, e. In the simulation below you can experiment with this system (which has been simulated using the Euler Integration Algorithm). The simulation shows a plot of x₁ against x₂ - a phase-plane plot. Here is the simulation.

Do the following:

Run the simulation with the values given for the starting value of x₁ and e. You should notice that the system exhibits a limit cycle. (Click here to go to limit cycle portion of the lesson on the phase plane where this system is also examined.)
Run the simulation using different starting values of x₁. Use values that put the starting point within the limit cycle you saw in the first part.
Run the simulation again using starting values that put the starting point outside the limit cycle you saw in the first part.
Compare the results. You should be able to see that the system goes to the same limit cycle behavior no matter where the system starts.

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